On Thursday, Christoph Aistleiter [from TU Gräz] gave a plenary talk at MCqMC 2016 around Hermann Weyl’s 1916 paper, Über die Gleichverteilung von Zahlen mod. Eins, which demonstrates that the sequence a, 22a, 32a, … mod 1 is uniformly distributed on the unit interval when a is irrational. Obviously, the notion was not introduced for simulation purposes, but the construction applies in this setting! At least in a theoretical sense. Since for instance the result that the sequence (a,a²,a³,…) mod 1 being uniformly distributed for almost all a’s has not yet found one realisation a. But a nice hour of history of mathematics and number theory: it is not that common we hear the Riemann zeta function mentioned in a simulation conference!

The following session was a nightmare in that I wanted to attend all four at once! I eventually chose the transport session, in particular because Xiao-Li advertised it at the end of my talk. The connection is that his warp bridge sampling technique provides a folding map between modes of a target. Using a mixture representation of the target and folding all components to a single distribution. Interestingly, this transformation does not require a partition and preserves the normalising constants [which has a side appeal for bridge sampling of course]. In a problem with an unknown number of modes, the technique could be completed by [our] folding in order to bring the unobserved modes into the support of the folded target. Looking forward the incoming paper! The last talk of this session was by Matti Vihola, connecting multi-level Monte Carlo and unbiased estimation à la Rhee and Glynn, paper that I missed when it got arXived last December.

The last session of the day was about probabilistic numerics. I have already discussed extensively about this approach to numerical integration, to the point of being invited to the NIPS workshop as a skeptic! But this was an interesting session, both with introductory aspects and with new ones from my viewpoint, especially Chris Oates’ description of a PN method for handling both integrand and integrating measure as being uncertain. Another arXival that went under my decidedly deficient radar.

In the recent days, we have had a lively discussion among AEs of the Annals of Statistics, as to whether or not set up a policy regarding publications of documents that have already been published in a shortened (8 pages) version in a machine learning conference like NIPS. Or AISTATS. While I obviously cannot disclose details here, the debate is quite interesting and may bring the machine learning and statistics communities closer if resolved in a certain way. My own and personal opinion on that matter is that what matters most is what’s best for Annals of Statistics rather than the authors’ tenure or the different standards in the machine learning community. If the submitted paper is based on a brilliant and novel idea that can appeal to a sufficiently wide part of the readership and if the maths support of that idea is strong enough, we should publish the paper. Whether or not an eight-page preliminary version has been previously published in a conference proceeding like NIPS does not seem particularly relevant to me, as I find those short papers mostly unreadable and hence do not read them. Since Annals of Statistics runs an anti-plagiarism software that is most likely efficient, blatant cases of duplications could be avoided. Of course, this does not solve all issues and papers with similar contents can and will end up being published. However, this is also the case for statistics journals and statistics, in the sense that brilliant ideas sometimes end up being split between two or three major journals.

Today I attended Persi Diaconis’ de Finetti’s ISBA Lecture and not only because I was an invited discussant, by all means!!! Persi was discussing his views on Bayesian numerical analysis. As already expressed in his 1988 paper. Which now appears as a foundational precursor to probabilistic numerics. And which is why I had a very easy time in preparing my discussion as I mostly borrowed from my NIPS slides. With some degree of legitimacy since I was already a discussant there. Anyway, here is the most novel slide in the discussion, built upon my realisation that the principle behind nested sampling is fairly generic for integral approximation, rather than being restricted to marginal likelihood approximation.

Among many interesting things, Persi’s talk made me think anew about infinite variance importance sampling. And about the paper by Souraj Chatterjee and Persi that I discussed a few months ago. In that some regularisation of those “useless” importance estimates can stem from prior modelling. Not as an aside, let me add I am very grateful to the ISBA 2016 organisers and to the chair of the de Finetti lecture committee for their invitation to discuss this talk!

Funny enough, I got a joking email from Brad, bemoaning my traitorous participation to the workshop on probabilistic numerics because of its “anti-MCMC” agenda, reflected in the summary:

“Integration is the central numerical operation required for Bayesian machine learning (in the form of marginalization and conditioning). Sampling algorithms still abound in this area, although it has long been known that Monte Carlo methods are fundamentally sub-optimal. The challenges for the development of better performing integration methods are mostly algorithmic. Moreover, recent algorithms have begun to outperform MCMC and its siblings, in wall-clock time, on realistic problems from machine learning.

The workshop will review the existing, by now quite strong, theoretical case against the use of random numbers for integration, discuss recent algorithmic developments, relationships between conceptual approaches, and highlight central research challenges going forward.”